Exact and approximate solutions of the convection-diffusion equation

Even in the case of laminar flows, very few analytic solutions of the convection-diffusion equation can be found in the literature. In the present investigation, it is shown that, at least in some cases, the convection-diffusion initial-value problem can be reduced to the simpler problem of solving a system of nonlinear ordinary differential equations. The employed procedure is based on a utilization of Lie algebra techniques, taking into account a proposal made by Wei and Norman (1963). Lie algebras are introduced with the aid of simple examples, including a model equation for an expanding universe with a time-dependent Hubble constant. The considered method is applied to unbounded circular and planar parabolic flow. For the case in which the flow is time dependent, the problem reduces to the solution of a Riccati equation. If the flow is time independent, the solution is given in terms of hyperbolic and trigonometric functions. Bounded flows are also studied.